The Dimensions of dimensionality

Cognitive scientists often infer multidimensional representations from data. Whether the data involve text, neuroimaging, neural networks, or human judgments, researchers frequently infer and analyze latent representational spaces (i.e., embeddings). However, the properties of a latent representation (e.g., prediction performance, interpretability, compactness) depend on the inference procedure, which can vary widely across endeavors. For example, dimensions are not always globally interpretable and the dimensionality of different embeddings may not be readily comparable. Moreover, the dichotomy between multidimensional spaces and purportedly richer representational formats, such as graph representations, is misleading. We review what the different notions of dimension in cognitive science imply for how these latent representations should be used and interpreted.